About an inequality of Kubota for plane convex figures (Q1591713)

This paper is concerned with the problem of finding the planar compact convex set with minimum area among all those with fixed diameter \(D\) and perimeter \(L\). If \(2D \leq L \leq 3D\) the optimal sets are well known (isosceles triangles), but the problem is still open for \(L\in (3D,\pi D)\). The author conjectures that in this last case the optimal sets have also maximal circumradius \(R=\frac D{\sqrt 3}\). So she studies the family of isoperimetric and isodiametric convex figures with maximal circumradius and obtains a new inequality which sharpens an inequality established by Kubota (1924). A certain kind of nonsymmetric inpolyeders of the Reuleaux triangles are the optimal sets. She also presents a complete system of inequalities concerning the perimeter, the diameter and the circumradius of a planar convex compact set.